Motion of System of Particles and Rigid Bodies (Detailed notes)

Center of Mass:

• Definition: Center of mass is the point where the entire mass of an object can be considered to be concentrated.
• Methods for determining the center of mass:
  • For symmetrical objects, the center of mass is at the geometrical center.
  • For irregularly shaped objects, the center of mass can be determined by taking moments about any axis.
  • For a system of particles, the center of mass is given by the formula:

$$ \bar{r} = \frac{\sum m_i \overrightarrow{r}_i}{M} $$

where ( \bar{r} ) is the position vector of the center of mass, (m_i) is the mass of the ( i)-th particle, (\overrightarrow{r_i}) is the position vector of the ( i)-th particle, and (M) is the total mass of the system.

• Reference: NCERT Physics Class 11, Chapter 7 - System of Particles and Rotational Motion

Linear Momentum and Conservation:

• Definition: Linear momentum is a vector quantity defined as the product of the mass of an object and its velocity.

• Conservation of Linear Momentum: The total linear momentum of an isolated system remains constant.

• Applications of linear momentum conservation:

• Solving collision problems -Analyzing rocket propulsion -Studying the recoil of firearms

• Reference: NCERT Physics Class 11, Chapter 7 - System of Particles and Rotational Motion

Impulse:

• Definition: Impulse is the product of force and the time interval over which it acts.
• Impulse-momentum theorem: The net impulse acting on an object is equal to the change in its momentum.

$$ \overrightarrow{J} = \overrightarrow{F} \Delta t = \Delta \overrightarrow{p}$$

• Applications of impulse-momentum theorem:

• Solving collision problems

• Analyzing the motion of rockets and projectiles

• Studying the recoil of firearms

• Reference: NCERT Physics Class 11, Chapter 7 - System of Particles and Rotational Motion

Elastic and Inelastic Collisions:

• Elastic collision: In an elastic collision, both momentum and kinetic energy are conserved.

• Inelastic collision: In an inelastic collision, momentum is conserved, but kinetic energy is not.

• Coefficient of restitution: A measure of the elasticity of a collision. $$ e = \frac{\text{relative velocity after collision}}{\text{relative velocity before collision}} = \frac{v_{21} - v_{12}}{v_{1i} - v_{2i}} $$

• Reference: NCERT Physics Class 11, Chapter 7 - System of Particles and Rotational Motion

Rotation and Torque:

• Angular displacement: The angle through which an object rotates about a fixed axis.

• Angular velocity: The rate at which an object rotates about a fixed axis. $$ \omega = \frac{\Delta \theta}{\Delta t} $$

• Angular acceleration: The rate at which an object’s angular velocity changes. $$ \alpha = \frac{\Delta \omega}{\Delta t} $$

• Torque: The rotational analogue of force. It is defined as the cross product of the force and the position vector of the point of application of the force with respect to the axis of rotation. $$\overrightarrow{\tau} = \overrightarrow{r} \times \overrightarrow{F}$$

• Reference: NCERT Physics Class 11, Chapter 7 - System of Particles and Rotational Motion

Moment of Inertia:

• Definition: Moment of inertia is a measure of an object’s resistance to rotational motion. It depends on the object’s mass distribution and its shape.

• Parallel axis theorem: The moment of inertia of an object about an axis parallel to its center of mass axis is equal to the moment of inertia about the center of mass axis plus the product of the object’s mass and the square of the distance between the two axes.

• Perpendicular axis theorem: The moment of inertia of an object about an axis perpendicular to one of its principal axes is equal to the sum of the moments of inertia about the other two principal axes.

• Reference: NCERT Physics Class 11, Chapter 7 - System of Particles and Rotational Motion

Rotational Dynamics:

• Torque-angular acceleration relation: The net torque acting on an object is equal to the product of its moment of inertia and its angular acceleration. $$\overrightarrow{\tau} = I \overrightarrow{\alpha}$$

• Rotational kinetic energy: The kinetic energy of a rotating object is equal to half of its moment of inertia times the square of its angular velocity. $$ K_r = \frac{1}{2} I\omega^2 $$

• Work done in rotating objects: The work done in rotating an object through an angle (\theta) is equal to the product of the net torque and the angle. $$ W = \overrightarrow{\tau} \cdot \overrightarrow{\theta} $$

• Reference: NCERT Physics Class 11, Chapter 7 - System of Particles and Rotational Motion

Angular Momentum and Conservation:

• Definition: Angular momentum is a vector quantity defined as the product of an object’s moment of inertia and its angular velocity.
• Conservation of Angular Momentum: The total angular momentum of